(Song et al., n.d.)
Background
Linear inverse problems
An inverse problem seeks to recover unobserved causal factors from a set of observed measurements.If \(x \in \mathbb{R}^n\) is an unknown signal and \(y \in \mathbb{R}^m=Ax+\epsilon\) is a noisy observation with \(m\) linear measurements and \(A\in \mathbb{R}^{m\times n}\) is a linear operator and \(\epsilon \in \mathbb{R^n}\) is a noise vector. Solving the inverse problem constitutes of recovering \(x\) from its measurement \(y\). There is the assumption that \(x\) is sampled from a prior \(p(x)\) and as such the measurement and signal are connected trough a the measurement distribution \(p(y|x)=q_{\epsilon}(y-Ax)\) with \(q_{\epsilon}\) being the noise distribution of \(\epsilon\). Given \(p(y|x)\) and \(p(x)\) the inverse problem can be solved by sampling from the posterior distribution \(p(x|y)\).
Score-based generative models
When solving an inverse problem we are given an observation \(y\), a measurement distribution \(p(y|x)\) and wish to sample from the posterior distribution \(p(x|y)\). The prior distribition \(p(x)\) is usually not known but we can use a generative model on a dataset $ {x_1,x_2 …x_n} p(x)$ to estimate the prior distribution. The posterior distribution \(p(x|y)\) can be determined through Bayes’ rule.